Mathematics
Grade 10
15 min
Identify hypotheses and conclusions
Identify hypotheses and conclusions
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Introduction & Learning Objectives
Learning Objectives
Define conditional statement, hypothesis, and conclusion.
Identify the hypothesis in a conditional statement.
Identify the conclusion in a conditional statement.
Rewrite sentences into the standard 'if-then' format.
Differentiate between a statement and its converse, inverse, and contrapositive.
Explain the role of hypotheses and conclusions in forming logical arguments for proofs.
If you finish your homework, then you can play video games. 🤔 What's the condition and what's the result?
This lesson introduces the building blocks of logical reasoning: conditional statements. We will learn how to break down these 'if-then' statements into their two core parts, the hypothesis and the conclusion. This skill is absolutely essenti...
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Key Concepts & Vocabulary
TermDefinitionExample
Conditional StatementA logical statement that has two parts, a hypothesis and a conclusion. It is typically written in 'if-then' form.If it is raining, then the ground is wet.
HypothesisThe 'if' part of a conditional statement. It is the condition or premise that is assumed to be true.In the statement 'If a polygon has three sides, then it is a triangle,' the hypothesis is 'a polygon has three sides'.
ConclusionThe 'then' part of a conditional statement. It is the result or outcome that follows from the hypothesis.In the statement 'If a polygon has three sides, then it is a triangle,' the conclusion is 'it is a triangle'.
NegationThe logical opposite of a statement. The symbol for negation is '...
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Core Formulas
Conditional Statement Form
p \rightarrow q
Represents 'If p, then q'. Here, 'p' is the hypothesis and 'q' is the conclusion.
Converse Form
q \rightarrow p
To find the converse, you switch the hypothesis (p) and the conclusion (q).
Inverse Form
\sim p \rightarrow \sim q
To find the inverse, you negate the original hypothesis (p) and negate the original conclusion (q). The '~' symbol means 'not'.
Contrapositive Form
\sim q \rightarrow \sim p
To find the contrapositive, you switch the hypothesis and conclusion AND negate them both.
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Challenging
A student rewrites the statement 'You must be at least 18 years old to vote' as 'If you are at least 18 years old, then you must vote.' What is the logical flaw in the student's conclusion?
A.The hypothesis is incorrect; it should be about voting.
B.The conclusion is too strong; being eligible to vote doesn't mean one is required to vote.
C.The student correctly identified the hypothesis but reversed the conclusion.
D.The statement cannot be written in if-then form.
Challenging
Consider the Pythagorean Theorem, which can be stated as: 'For any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.' What is the hypothesis of this theorem when written in if-then form?
A.triangle is a right triangle.
B.The square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
C.The lengths of the two sides are squared.
D.The triangle has a hypotenuse.
Challenging
Given the conditional statement: 'If a figure is a rhombus, then its diagonals are perpendicular bisectors of each other.' Which of the following statements is logically equivalent to the original statement?
A.If a figure's diagonals are perpendicular bisectors, then it is a rhombus.
B.If a figure is not a rhombus, then its diagonals are not perpendicular bisectors.
C.If a figure's diagonals are not perpendicular bisectors, then it is not a rhombus.
D.figure is a rhombus if and only if its diagonals are perpendicular bisectors.
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